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An article in Babbage's Bag is all about chaos and this seems like a good time to have some fun plotting some chaos diagrams and using the scrolling technique described in the recent project Stripchart Scrolling.
The basic idea is very simple. The iteration
x<ax(1x)
is performed to see what values of x are generated for various values of the constant a.
The values of a that are of interest are between about 2.95 and 4.00. For this range the iteration stays within the interval 0 to 1 and chaos sets in at about a = 3.5.
When the iteration is started it takes some time to settle down and then it cycles though a finite number of values. The values that the iteration cycles though form the attracting set of the process and it’s these we need to plot.
Start a new C# project and place a button and a picturebox on the form.
The form you need to show chaos
The button’s click routine does the work of plotting. First we need to make the pictureBox have a persistent drawing surface in the form of a bitmap:
private void button1_Click( object sender, EventArgs e) { Bitmap BImage1 = new Bitmap( pictureBox1.ClientSize.Width, pictureBox1.ClientSize.Height); pictureBox1.BackgroundImage = BImage1;
Next we work out a scaling that gives the correct range of values for to cover the entire picturebox:
Double a; Graphics G = pictureBox2.CreateGraphics(); for (int y = 0; y < BImage1.Height; y++) { a = 2.95 + (4  2.95) * y / BImage1.Height;
We then initialise x to some convenient value, the eventual result doesn’t depend much on what this is, and then iterate 500 times to allow transients to die away:
Double x = 0.2; for(int i = 1;i<500;i++) { x = a * x * (1  x); }
The next 500 iterations are used to find out what values of x occur in the iteration after the transients have gone:
for ( int i= 1; i < 500; i++) { x = a * x * (1  x); BImage1.SetPixel( (int)(x * BImage1.Width), y,Color.Green); int y1 =(int) (x * pictureBox2.Height); G.DrawLine(Pens.Red, 0, y1, 0, y1 + 1); pictureBox2.Scroll(1, 0); Application.DoEvents(); }
Finally we Refresh the pictureBox and make sure to dispose of the Graphics object:
pictureBox1.Refresh(); } G.Dispose(); }
If you run this finished program you will see the bifurcation diagram for the map  see the next page.
<ASIN:0738204536>
<ASIN:0387202293>
<ASIN:0821801376>
<ASIN:1559533560>
